This part of the project comprises 4 tasks: 1a, 1b, 1c , 2. Task 1b counts as much as the three other tasks together.
General performance of the groups:
Task 1a: This was mostly solved correctly. Some groups managed the first subtask (showing that the slution of the given matrix differential equation is an orthogonal matrix) but not the last part (showing that the function Φ decreases along sloutions of the differential equation).
Task 1b: This was the most rewarding task. Morsly the signal separation was successfull, the ICA technique was implemented correctly to a large extent among the groups. Also the forward Euler method was implemented correctly. However, some people implemented the projected backward Euler wrongly, they did not realize that the projection is part of the method. The method you were asked to implement is not obtained by performing a step of the backward Euler and than performing the projection, but rather doing all at once. This means that the projection (QR decomposition) should be within each step of the fixed point iteration, and not added after the iteration is concluded. Many groups did not report information on the number of iterations used by their codes in the fixed point iteration in the implementation of the backward Euler method, and did not use a tolerance to stop the iteration. Some used a fixed (and rather low) number of iterations, so that there is no guarantee that the fixpoint iteration has converged. These missing details are lowering the quality of the reports, and leaving the reader with a sense of unpleasant uncertainty. Some groups presented very informative tables and figures.
Task 1c: Many were creative here and did a very good job. Some groups really showed that they had understood properly the ICA technique and could comment properly on the advantages and limitation of this procedure to separate signals. Most of you used interesting data and really illustrated that this method works if all the assumptions are fullfilled.
Problem 2: the first part is very standart one should just apply the theorem on the interpolation error from the book in the case of an interpolation polynomial of degree 1. The second part should be handled with care and it tricked many of you: if f(x) is a quadratic polynomial f(x)=ax^2+bx+c then you get equality (this includes also the case where a=0). Other choices of functions typically do not achieve equality in the bound.